Questions: Path Difference and Constructive/Destructive Interference

Question 1

A listener is 3.0 m from one speaker and 4.0 m from a second speaker. Both emit sound with a wavelength of 1.0 m. What type of interference does the listener experience?

• A. Constructive, because the path difference is 1.0 m, which equals one full wavelength
• B. Destructive, because the average distance of 3.5 m is not a whole-number multiple of λ
• C. Constructive, because the nearest speaker is only 3.0 m away, which is 3λ
• D. Destructive, because the path difference of 1.0 m is less than the distance to either source

Answer: (A)

Path difference Δ = 4.0 − 3.0 = 1.0 m = 1λ. Since Δ = nλ (with n = 1), constructive interference occurs. The critical mistake in options B–D is using individual distances or averages instead of the difference. Only the difference between the two path lengths determines whether waves arrive in phase or out of phase.

Question 2

Two coherent wave sources produce waves with wavelength 0.4 m. At a detector, the path difference is 0.6 m. What is the interference condition?

• A. Destructive, because 0.6 m = 1.5λ, satisfying Δ = (n + ½)λ with n = 1
• B. Constructive, because 0.6 m is less than one full wavelength
• C. Constructive, because 0.6 m ≈ 1.5λ, which is 'close enough' to a whole-number multiple
• D. Indeterminate — only Δ = 0 guarantees a predictable interference condition

Answer: (A)

0.6 m / 0.4 m = 1.5, so Δ = 1.5λ = (1 + ½)λ, the destructive interference condition. Option B confuses magnitude with the interference criterion — what matters is not how large Δ is, but whether it equals nλ or (n + ½)λ. Option C is wrong because 'close enough' has no meaning here; the condition must be satisfied exactly for perfect cancellation.

Question 3

Constructive interference can occur at a point equidistant from both sources.

• T. True
• F. False

Answer: True

At any point equidistant from both sources, the path difference Δ = 0, which satisfies Δ = nλ with n = 0. This is the zeroth-order constructive maximum — the central bright fringe in a double-slit experiment. Zero path difference means the two waves are perfectly in phase because neither has traveled any extra distance.

Question 4

The path difference at any given point equals the distance from the nearer source to that point.

• T. True
• F. False

Answer: False

Path difference is the *difference* between the two path lengths, not either path length alone. If one source is 3 m away and the other is 5 m away, the path difference is 2 m — not 3 m or 5 m. Confusing path difference with a single distance is the most common error in interference problems, and it leads to completely wrong predictions about which points are bright or dark.

Question 5

Why is it the *difference* in path lengths — rather than the individual path lengths themselves — that determines whether interference is constructive or destructive?

Model answer: Two coherent sources start in phase. Any extra distance one wave travels corresponds to extra cycles it completes before reaching the detector. If the extra distance (the path difference) is a whole number of wavelengths, the wave arrives having completed full extra cycles and is back in phase — constructive. If the extra distance is a half-plus-integer number of wavelengths, it arrives exactly half a cycle off — destructive. The absolute distance from either source doesn't matter because it contributes equally to both waves' phase; only the *difference* shifts one wave relative to the other.

Phase is set by how many wavelengths fit into the path. Since both waves start in phase, what matters is whether the extra path of the farther wave corresponds to a whole number of wavelengths (back in phase) or a half-integer number (inverted). The individual distances only matter insofar as their difference reveals this extra cycle count.